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Mathematical Interests

I am a Professor and Rosemary Kopel Brown Eminent Scholars Chair in the Mathematics Department at Auburn University. My research interests are interdisciplinary: my background is in commutative algebra and algebraic geometry. I'm especially interested in problems which can be studied from a computational standpoint, and in interactions with problems in applied mathematics. My current applied work is on:

On the more theoretical side, I enjoy working on problems at the interface of discrete geometry and algebra; for example I've also worked on questions involving free resolutions and syzygies, coding theory, rational surfaces and postulation, stability and jump loci of vector bundles, and geometric complexity theory. I did my graduate work at Cornell, and postdoctoral work at Cornell, Harvard, and Northeastern. Prior to coming to Auburn I served as Chair at Iowa State, and I've also been a professor at Illinois and Texas A&M; visiting positions include a Leverhulme Professorship in Oxford and a Fulbright Professorship in Buenos Aires. Complete vita in html and pdf.
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Information for Students

In fall 2023 I'm teaching Calculus III. Past classes I've taught at Auburn include Linear Algebra, Abstract Algebra II, Calculus III, Computational Algebra, and Topological Data Analysis. For a cool application of linear algebra, here is a link to a minicourse I taught for talented high school students on the mathematics of Google. For students interested in commutative algebra and algebraic geometry, here are a few thoughts on what to read and how to choose an advisor. Below are webpages of students and postdocs:
Here are notes from a lunch talk I've given several times on the Job Hunt.
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Books

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Upcoming/Recent Talks and Conferences

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Conference and Workshop Organization

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Publications

o Betti tables forcing failure of the Weak Lefschetz Property (with Sean Grate), The strong and weak Lefschetz properties, Springer INdAM series, to appear.
o Free resolutions and Lefschetz properties of some Artin Gorenstein rings of codimension four (with Nancy Abdallah), Journal of Symbolic Computation, to appear.
o Calabi-Yau threefolds in P^n and Gorenstein rings (with Mike Stillman and Beihui Yuan), Advances in Theoretical and Mathematical Physics, 26, (2022), 764-792.
o Nets in P^2 and Alexander Duality (with Nancy Abdallah), Discrete and Computational Geometry, to appear.
o Algebraic properties of Hermitian sums of squares (with Jennifer Brooks and Dusty Grundmeier), Proceedings of the A.M.S., 150, (2022), 3471-3476.
o Schemes supported on the singular locus of a hyperplane arrangement in P^n (with Juan Migliore and Uwe Nagel), International Mathematics Research Notices, 1, (2022), 140-170.
o The simplest minimal free resolutions in P^1 x P^1, (with Nicolas Botbol and Alicia Dickenstein), in Commutative Algebra, Springer Verlag (2021), 113-145.
o The Hessian polynomial and the Jacobian ideal of a reduced hypersurface in P^n, (with Laurent Buse, Alexandru Dimca, and Gabriel Sticlaru), Advances in Mathematics, 392, (2021), 22 pages.
o Quadratic Gorenstein rings and the Koszul property I (with Matt Mastroeni and Mike Stillman), Transactions of the A.M.S., 374, (2021), 1077-1093.
o Quadratic Gorenstein rings and the Koszul property II, (with Matt Mastroeni and Mike Stillman), International Mathematics Research Notices, 2, (2023), 1461-1482.
o Rees Algebras, Syzygies, and Geometric Modeling, in Applications of polynomial systems, by David Cox, CBMS Regional Conference Series in Mathematics 134, (2020), 121-135.
o The Weak Lefschetz property for quotients by Quadratic Monomials (with Juan Migliore and Uwe Nagel), Mathematica Scandinavica, 126, (2020), 41-61.
o Trading Networks and Hodge Theory (with Richard Sowers and Rui Song), Journal of Physics Communications, (2020), 21pp.
o A new bound for smooth spline spaces (with Mike Stillman and Beihui Yuan), Journal of Combinatorial Algebra, 4, (2020), 359-367.
o Stratifying multiparameter persistent homology (with Heather Harrington, Nina Otter and Ulrike Tillmann), SIAM Journal on Applied Algebra and Geometry, 3, (2019), 439-471.
o Logarithmic vector fields for curve configurations in P^2 with quasihomogeneous singularities, (with Hiroaki Terao and Masahiko Yoshinaga), Mathematical Research Letters, 25, (2018), 1977-1992.
o The method of shifted partial derivatives cannot separate the permanent from the determinant (with Klim Efremenko, J.M. Landsberg, Jerzy Weyman), Mathematics of Computation, 87, (2018), 2037-2045.
o On minimal free resolutions of subpermanents and other ideals arising in complexity theory (with Klim Efremenko, J.M. Landsberg, Jerzy Weyman), Journal of Algebra, 503, (2018), 8-20.
o Subdivision and spline spaces (with Tatyana Sorokina), Constructive Approximation, 47, (2018), 237-247.
o Codes from surfaces with small Picard number, (with John Little), SIAM Journal on Applied Algebra and Geometry, 2, (2018), 242-258.
o Polynomial interpolation in higher dimension: from simplicial complexes to GC sets (with Nathan Fieldsteel), SIAM Journal on Numerical Analysis, 55 (2017), 131-143.
o Algebraic methods in Approximation theory, Computer Aided Geometric Design, 45 (2016), 14-31.
o Tensor product surfaces and linear syzygies (with Eliana Duarte), Proceedings of the A.M.S., 144 (2016), 65-72.
o Chen ranks and resonance, (with Dan Cohen), Advances in Mathematics, 285 (2015), 1-27.
o Finitely many smooth d-polytopes with n lattice points, (with Bogart, Haase, Hering, Lorenz, Nill, Paffenholz, Rote, Santos), Israel Journal of Mathematics, 207, (2015), 301-330.
o Geometry of Wachspress surfaces, (with Corey Irving), Algebra & Number Theory, 8 (2014), 369-396.
o Splines on the Alfeld split of a simplex and type A root systems, Journal of Approximation Theory, 182 (2014), 1-6.
o Syzygies and singularities of tensor product surfaces of bidgree (2,1), (with Alexandra Seceleanu and Javid Validashti), Mathematics of Computation, 83 (2014), 1337-1372.
o Local cohomology of logarithmic forms, (with Graham Denham, Matthias Schulze, Max Wakefield, Uli Walther), Annales de l'Institut Fourier, 63 (2013), 1177-1203.
o Commutative algebra of subspace and hyperplane arrangements, (with Jessica Sidman), in Commutative Algebra, Springer Verlag (2013), 639-665.
o Toric Hirzebruch-Riemann-Roch via Ishida's theorem on the Todd genus, Proceedings of the A.M.S., 141 (2013), 1215-1217.
o High rank linear syzygies on low rank quadrics, (with Mike Stillman), American Journal of Mathematics, 134 (2012), 561-579.
o Equivariant Chow cohomology of nonsimplicial toric varieties, in Transactions of the A.M.S., 364 (2012) 4041-4051.
o Hyperplane Arrangements: Computations and Conjectures, in Advanced Studies in Pure Mathematics, 62, (2012) 323-358.
o Recent developments and open problems in linear series, (with the MFO miniworkshop crew), Contributions to Algebraic Geometry (P. Pragacz, ed.) EMS publishing (2012) 93-140.
o Resonance varieties via blowups of P^2 and scrolls, in International Mathematics Research Notices, 20, (2011) 4756-4778.
o Inverse systems, Gelfand-Tsetlin patterns and the weak Lefschetz property, (with Brian Harbourne, Alexandra Seceleanu), in Journal of the London Mathematical Society, 84, (2011) 712-730.
o Euler characteristic of coherent sheaves on simplicial torics via the Stanley-Reisner ring, in Journal of Mathematical Physics, 51, (2010) doi 112304.
o The weak Lefschetz property and powers of linear forms in K[x,y,z], (with Alexandra Seceleanu), in Proceedings of the A.M.S., 138 (2010) 2335-2339.
o Freeness of Conic-Line arrangements in P^2, (with Stefan Tohaneanu), in Commentarii Mathematici Helvetici, 84 (2009) 235-258.
o Holonomy Lie algebras and the LCS formula for subarrangements of A_n, (with Paulo Lima-Filho), International Mathematics Research Notices, 8 (2009) 1421-1432.
o Piecewise polynomials on polyhedral complexes, (with Terry Mcdonald), Advances in Applied Mathematics, 42 (2009), 82-93.
o The Orlik-Terao algebra and 2-formality, (with Stefan Tohaneanu), Mathematical Research Letters, 16 (2009), 171-182.
o Efficient computation of resonance varieties via Grassmannians, (with Paulo Lima-Filho), Journal of Pure and Applied Algebra, 213 (2009), 1606-1611.
o A case study in bigraded commutative algebra, (with David Cox and Alicia Dickenstein), in "Syzygies and Hilbert Functions", edited by Irena Peeva, Lecture notes in Pure and Applied Mathematics 254, (2007), 67--112.
o Betti numbers and degree bounds for some linked zero schemes, (with Leah Gold and Hema Srinivasan), Journal of Pure and Applied Algebra, 210, (2007), 481-491.
o Syzygies, multigraded regularity and toric varieties, (with Milena Hering and Gregory G. Smith), Compositio Mathematica, 142, (2006), 1499-1506.
o Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence, (with Alex Suciu), in Transactions of the A.M.S., 358, (2006), 2269-2289.
o Toric surface codes and Minkowski sums, (with John Little), SIAM Journal on Discrete Mathematics, 20, (2006), 999-1014.
o Derivation modules of orthogonal duals of hyperplane arrangements, (with Joseph P.S. Kung), in Journal of Algebraic Combinatorics, 24, (2006), 253-262.
o Cayley-Bacharach and evaluation codes on complete intersections, (with Leah Gold and John Little), in Journal of Pure and Applied Algebra, 196, (2005) p. 91-99.
o Linear systems on a special rational surface, in Mathematical Research Letters, 11, (2004) p. 697-714.
o Lattice polygons and Green's theorem, in Proceedings of the A.M.S., 132, (2004) p. 3509-3512.
o Elementary modifications and line configurations in P^2, in Commentarii Mathematici Helvetici, 78, (2003) p. 447-462.
o Local complete intersections in P^2 and Koszul syzygies, (with David Cox), in Proceedings of the A.M.S., 131, (2003) p. 2007-2014.
o Lower central series and free resolutions of hyperplane arrangements, (with Alex Suciu), in Transactions of the A.M.S., 354, (2002) p. 3409-3433.
o Cohomology vanishing and a problem in approximation theory, (with Peter Stiller), in Manuscripta Mathematica, 107, (2002) p. 43-58.
o The module of logarithmic p-forms of a locally free arrangement, (with Mircea Mustata), in Journal of Algebra, 241, (2001) p. 699-719.
o On a conjecture of Rose, (with John Dalbec), in Journal of Pure and Applied Algebra, 165, (2001) p. 151-154.
o A rank two vector bundle associated to a three arrangement, and its Chern polynomial, in Advances in Mathematics, 149, (2000) p. 214-229.
o Subalgebras of the Stanley-Reisner ring, in Discrete and Computational Geometry, 21, (1999) p. 551-556.
o Fat points, inverse systems, and piecewise polynomial functions, (with Anthony Geramita), in Journal of Algebra, 204, (1998) p. 116-128.
o A spectral sequence for splines, in Advances in Applied Mathematics, 19, (1997) p. 183-199.
o Local cohomology of bivariate splines, (with Mike Stillman), in Journal of Pure and Applied Algebra, 117-118, (1997) p. 535-548.
o A family of ideals of minimal regularity and the Hilbert series of C^r(\Delta), (with Mike Stillman), in Advances in Applied Mathematics, 19, (1997) p. 169-182.
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Preprints

o Free curves, Eigenschemes, and Pencils of curves (with Roberta Di Gennaro, Giovanna Illardi, Rosa Maria Miro-Roig, and Jean Valles).
o Quaternary quartic forms and Gorenstein rings (with Grzegorz Kapustka, Michal Kapustka, Kristian Ranestad, Mike Stillman and Beihui Yuan).
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Computational Mathematics

oMacaulay2: A program for computing in commutative algebra and algebraic geometry.
oCoCoA: Another option for computing in commutative algebra and algebraic geometry.
oPorta (and others): Programs for polyhedral geometry.
oSingular: A program for computations in local rings and singularities.
oSchubert: A program for enumerative geometry.
oSplinetool: Macaulay2 script which builds the chain complexes and allows computation of various homology modules associated to splines.
oArrangetool: Macaulay2 script for computations involving three arrangements.
oMPH: Macaulay2 scripts for computations involving multiparameter persistent homology.
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Useful Math Sites

o Association for Mathematical Research
o American Math Society
o Society for Industrial and Applied Math
o Algebraic Geometry preprints
oMSRI (Berkeley).
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